Disturbing the fiscal theory of the price level: Can it fit the eu-15?

this allows us to write

P

1 + i_t = (1 + r_t ) -ɪ .                             (19)

P_t

Bringing to mind the situation mentioned above, where the monetary base growth rate is
constant, and substituting the inverse of (19) in (16), results in

P1P

b_t+1 — —--- = b + g_t - tx_t                 (20)

^t+1 P_t 1 + r_tP_t+ι     ^t

b_t+1 = (1 + r_t )b_t + (1 + r_t )(gt - txt ).                       (21)

Assuming also the hypothesis of a constant real interest rate, and for simplicity, if the
budget deficit is stable, (g_t - tx_t ) = (g - tx), we have also

bt+1 = (1 + r)bt + (1 + r )(g - tx).                         (22)

From the the last expression, it becomes clear that bt = Bt/Pt will follow an explosive
trajectory since (1+r)>0. Notice also that from equation (22), the growth rate of
government debt is given by the following difference equation

bs

ɪ = (1 + r )(1 - _ ),                       (23)

b_t                  b_t

where the primary budget surplus, _, is given by _ = tx-g, and that eventually (23)
converges to (1+r) while b increases. However, in this case, the government is
conducting Ponzi games, and it would no be possible to satisfy a transversality
condition such as the one given by equation (7).

An explosive situation for the stock of real government debt will be avoided if the initial
value for b is

^b0 =^{-(1 + r)(}g ^{- tx})/ ^{r ,                          (24)}

in order to ensure that b remains constant at that same value. As a matter of fact, with
that initial value for b one gets simply

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